Analysis on $\bR^n$

What does analysis on $\bR^n$ -mean-?
You can do analysis on any manifold
  (some analysis on any measure space,
   and $p$-adic analysis on $p$-adic manifolds)

- analysis on $[0,1]$ or $(-1,1)$ is -local- structure
- analysis on a -group- is interesting
  (Fourier theory/representation theory)
- but $\bR^n$ shows up more deeply as the -tangent- (or -cotangent-)
  space -- so analysis on $\bR^n$ tells you about the -infintesimal-
  structure

Thus:
- infintesimal: line/space, $\bR^n$
- local: interval, $(-1,1)$ or $[0,1]$
- global: on a group or manifold: $G$ or $M$

The global question generally becomes topological very quickly,
or dynamics (b/c that's the key way for the structure of the manifold
to matter analytically).
